noboru wrote:
If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?
I. \(\frac{\sqrt{x+y}}{2}\)
II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)
III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)
A. I only
B. II only
C. III only
D. I and II only
E. None
Let's
test some values.x = 1 and y = 11/√(x + y) = 1/√(1 + 1) =
1/√2I. √(x + y)/2 = √(1 + 1)/2 = √2/2
Notice that, if we take
1/√2 and multiply top and bottom by √2, we get: √2/2, which is the same as quantity I
Since quantity I is not greater than
1/√2,
statement I is not trueII. (√x + √y)/2 = (√1 + √1)/2 = (1 + 1)/2 = 2/2 = 1
Since 1 IS greater than
1/√2, we cannot say for certain whether quantity II will
always be greater than √(x + y)/2
III. (√x - √y)/(x + y) = (√1 - √1)/(1 + 1) = (1 - 1)/2 = 0/2 = 0
Since 0 is not greater than
1/√2,
statement III is not trueSo, statements I and III are definitely not true, and we aren't yet 100% certain about statement II
Let's try another pair of values for x and y
x = 0.25 and y = 0.251/√(x + y) = 1/√(0.25 + 0.25) =
1/√0.5Let's further simplify
1/√0.5Since 1 = √1, we can say:
√1/√0.5Then we'll use a rule that says (√k)/(√j) = √(k/j)
So,
√1/√0.5 = √(1/0.5) = √2We see that, when x = 0.25 and y = 0.25, 1/√(x + y) =
√2II. (√x + √y)/2 = (√0.25 + √0.25)/2 = (0.5 + 0.5)/2 = 1/2
Since 1/2 is NOT greater than
√2,
statement II is not trueAnswer:
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